Abstract : An investigation was made to determine whether or not a certain map from universal Teichmuller space, T, to bounded operators in a certain Banach space was a homomorphism. It is found that this map is not homomorphic but, at the same time, precisely the extent to which it fails to be homomorphic. The other chief results of the paper are the theorems 1 through 5. Theorems 1 and 2 give two descriptions of right translation which are, in a sense, dual to each other. Theorem 3 is a local description of the derivative of right translation on the tangent bundle of Teichmuller space. Its importance is that it yields a corollary which gives new reproducing formulas and it makes possible theorem 5 which gives an explicit form of the operator 1/L(2, C to the mu power). Theorem 4 gives a geometric method of constructing inverses in Teichmuller space when points in T are viewed as quasicircles passing through 0, 1, and infinity. Namely, the inverse is given simply by complex conjugation. (Author)